# Definition:Vector Field

## Definition

Let $F$ be a field which acts on a region of space $S$.

Let the point-function giving rise to $F$ be a vector quantity.

Then $F$ is a **vector field**.

## Classification

### Conservative Vector Field

$\mathbf V$ is a **conservative (vector) field** if and only if its curl is everywhere zero:

- $\curl \mathbf V = \bszero$

### Solenoidal Vector Field

$\mathbf V$ is defined as being **solenoidal** if and only if its divergence is everywhere zero:

- $\operatorname {div} \mathbf V = 0$

## Examples

### Velocity of Fluid

In a moving fluid, the velocity $\mathbf v$ of the fluid is an example of a vector field.

That is, the velocity $\mathbf v$ at a point $P$ in the fluid is the velocity of the particle which is situated at $P$ at a given instant.

### Electric Field Strength

Let $R$ be a region of space in which there exists an electric field.

The electric field strength in $R$ gives rise to a vector field over $R$.

### Magnetic Field Strength

Let $R$ be a region of space in which there exists an magnetic field.

The magnetic field strength in $R$ gives rise to a vector field over $R$.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $5$. Scalar and Vector Fields - 1961: D.R. Bland:
*Solutions of Laplace's Equation*... (next): Chapter $1$: Occurrence and Derivation of Laplace's Equation: $1$. Situations in which Laplace's equation arises.